Iterative Solver for Linear System Obtained by Edge Element: Variable Preconditioned Method With Mixed Precision on GPU

The variable preconditioned (VP) Krylov subspace method with mixed precision is implemented on graphics processing unit (GPU) using compute unified device architecture (CUDA), and the linear system obtained from the edge element is solved by means of the method. The VPGCR method has the sufficient condition for the convergence. This sufficient condition leads us that the residual equation for the preconditioned procedure of VPGCR can be solved in the range of single precision. To stretch the sufficient condition, we propose the hybrid scheme of VP Krylov subspace method that uses single and double precision operations. The results of computations show that VPCG with mixed precision on GPU demonstrated significant achievement than that of CPU. Especially, VPCG-JOR on GPU with mixed precision is 41.853 times faster than that of VPCG-CG on CPU.

[1]  Hajime Igarashi,et al.  Convergence of preconditioned conjugate gradient method applied to driven microwave problems , 2003 .

[2]  K. Fujiwara,et al.  Acceleration of Convergence Characteristic of Iccg Method , 1992, Digest of the Fifth Biennial IEEE Conference on Electromagnetic Field Computation.

[3]  Y. Takahashi,et al.  Folded Preconditioner: A New Class of Preconditioners for Krylov Subspace Methods to Solve Redundancy-Reduced Linear Systems of Equations , 2009, IEEE Transactions on Magnetics.

[4]  John D. Owens,et al.  General Purpose Computation on Graphics Hardware , 2005, IEEE Visualization.

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  Attila Kakay,et al.  Speedup of FEM Micromagnetic Simulations With Graphical Processing Units , 2010, IEEE Transactions on Magnetics.

[7]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[8]  Hajime Igarashi,et al.  On the property of the curl-curl matrix in finite element analysis with edge elements , 2001 .

[9]  Koji Fujiwara,et al.  Improvement of convergence characteristic of ICCG method for the A-/spl phi/ method using edge elements , 1996 .

[10]  Koji Fujiwara,et al.  Improvement of convergence characteristic of ICCG method for the A-φ method using edge elements , 2010 .

[11]  Kuniyoshi Abe,et al.  A VARIABLE PRECONDITIONING USING THE SOR METHOD FOR GCR-LIKE METHODS , 2005 .

[12]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[13]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[14]  M. Shimasaki,et al.  A Novel Algebraic Multigrid Preconditioning for Large-Scale Edge-Element Analyses , 2006, 2006 12th Biennial IEEE Conference on Electromagnetic Field Computation.

[15]  H. Igarashi,et al.  Effect of Preconditioning in Edge-Based Finite-Element Method , 2008, IEEE Transactions on Magnetics.

[16]  Soichiro Ikuno,et al.  Numerical Investigations of Variable Preconditioned GCR with Mixed Precision on GPU , 2010 .